Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
136 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C251 | #106 |
| commutator | C8 | C42×C251 |
| Frattini, cosocle | C22×C8 | C22×C251 |
| prime radical | C23×C251 | SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C251 ◁ C22×C4×C251 ◁ C22×C8×C251 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 9. C2×C16×C251 | C4 | semidirect |
| 11. #108×C251 | C4 | semidirect |
| 55. C251 | #1383 | direct (N × H) |
| 56. #1383 | C251 | direct (N × H) |
| 65. C2×C16 | C4×C251 | semidirect |
| 67. #108 | C4×C251 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×M32×C251 | 2 |
| C2×#108×C251 | 2 |
| C2×#106×C251 | 2 |
| #1383 | 251 |